Method: Mind the GapWebinar Series

Overview of StatisticalModels for the Design

and Analysis of Stepped Wedge Cluster Randomized Trials

Presented by:Fan Li, Ph.D. Yale University School of Public Health NIH, National Institutes of H ealth

Office of Disease Prevention

If you’re a person with a disability or using assistive technology and are having difficulty accessing any of the content in this file, please contact Astrid Masfar at astrid.masfar@nih.gov.

Overview of Statistical Models for the Design and Analysis of Stepped Wedge

Cluster Randomized Trials

Fan Li

Assistant Professor Department of Biostatistics, and

Center for Methods in Implementation and Prevention Science Yale University School of Public Health

Offce of Disease Prevention: Mind the Gap Seminar Series

July 14, 2020

1\57

Acknowledgement I NIH Collaboratory Biostatistics and Study Design Core

Working Group I Drs. David Murray & Patrick Heagerty for discussions

leading up to this article as part of the work from the working group

I Collaborators on the SMMR article1

I James Hughes (UW), Patrick Heagerty (UW), Edward Melnick (Yale), Karla Hemming (UBir), Monica Taljaard (OHRI)

I Other collaborators I John Preisser (UNC), Elizabeth Turner (Duke), Paul

Rathouz (UT Austin), Donna Spiegelman (Yale), Xin Zhou (Yale), Hengshi Yu (Umich), Zibo Tian (Yale), Jiachen Chen (Yale) . . .

1Li F. et al (2020) Mixed-effects models for the design and analysis of stepped wedge cluster randomized trials: An overview. Stat Methods Med Res 2\57

Funding Acknowledgement

Supported by

I Administrative supplement 3-UH3-DA047003-02S2 from the NIH Offce of Disease Prevention & National Institute on Drug Abuse

I PCORI award ME-1507-31750 (PI: Heagerty)

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1. Introduction

4\57

Introduction I Cluster randomized trials (CRTs) allocate clusters of

individuals to intervention

I minimize contamination

I administrative convenience

I usually in parallel design

I Stepped wedge (SW) design rolls out intervention in a staggered fashion

I logistical constraints

I perceived ethical beneft

I Other pros and cons discussed extensively by Hemming and Taljaard2

2Hemming, K., Taljaard, M. (2020). Refection on modern methods: when is a stepped-wedge cluster randomized trial a good study design choice?. Int. J. Epidemiol.. 5\57

Analytical Models

I Unique features requires of SW designs require more complex considerations on analytical models

I Mixed-effects models I seminal methods paper of Hussey and Hughes (2007)3

I fixed-effects for time & intervention

I random-effects for clustering

I Among other modeling alternatives, mixed-effects models are more accessible from standard software, and are most widely used in SW-CRTs (Barker et al. 2016, BMC Med. Res. Methodol.)

3Hussey MA, Hughes JP (2007) Design and analysis of stepped wedge cluster randomized trials. Contemp Clin Trials 6\57

Analytical Models - Cont’d

I Many extensions of basic model over past decade

I Systematic reviews (Davey et al. 2015; Martin et al. 2016; Barker et al. 2016; Grayling et al. 2017)

I statistical methods for the sample size determination varied across studies

I insufficient details on modeling assumptions were provided

I reproducibility and sensitivity

I Integrate the toolkit of analytical models for SW-CRTs

I essential ingredients?

I common variants?

I identify areas that need further development or assessment

7\57

CONSORT extension to SW-CRTs I CONSORT item 7a: Sample size4

I Extension for SW-CRTs – . . . Method of calculation and relevant parameters with sufficient detail so the calculation can be replicated. Assumptions made about correlations between outcomes of participants from the same cluster.

I CONSORT item 12a & b: Statistical methods

I Extension for SW-CRTs – . . . Statistical methods used to compare treatment conditions for primary and secondary outcomes including how time effects, clustering, and repeated measures were taken into account..

I High-level, general model representation

I introduce model variants

I clarify assumptions and implications 4Hemming K (2018) Reporting of stepped wedge cluster randomised

trials: Extension of the CONSORT 2010 statement with explanation and elaboration. BMJ 8\57

2. General Model Representation

9\57

Terminology I Consider a complete stepped wedge CRT with I

participating clusters followed over J (J ≥ 3) time periods

I Cross-sectional (CS)

I different individuals observed in each cluster over time

I assume Nij individuals are included during period j in cluster i

I Closed-cohort (CC)

I individuals identifed at the start of the trial and scheduled for repeated outcome assessment

I Ni as the cohort size in cluster i as repeated measurements are taken from the same individuals

I Open-cohort (OC): a mix of the two

I Cluster starts out in the control; sets of clusters randomized to intervention until all clusters exposed

10\57

Cluster-Period Diagram j = 1 j = 2 j = 3 j = 4 j = 5

i = 4

i = 3

i = 2

i = 1

i = 8

i = 7

i = 6

i = 5

I schematic illustration with I = 8 clusters and J = 5 periods. Each white cell indicates a cluster-period under the control condition and each gray cell indicates a cluster-period under the intervention condition. There are in total S = 4 distinct intervention sequences

I each one of the 4 distinct intervention sequences is fully determined by the time period during which the intervention is frst implemented

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Outcome Model I Yijk (s): the outcome of individual k during period j in

cluster i , had cluster i received an intervention sequence s

I Mean model

g[µijk (s)] = Fi (j , s)0θ + Rik (j , s)0αi ,

I µijk (s) conditional mean of Yijk (s), g link function

I Fi (j , s)0θ: the group-average outcome trajectory and θ includes the parameter of interest (i.e., the intervention effect)

I Rik (j , s)0αi : the cluster-specific, time-specific and/or individual-specific departure from the group average

I A GLM, but borrow the “potential outcome" language toclearly indicate the dependence of elements on intervention sequence s5

5Sitlani CM, Heagerty PJ et al. (2012) Longitudinal structural mixed models for the analysis of surgical trials with noncompliance. Stat Med 12\57

Outcome Model - Cont’d

I Separate Fi (j , s)0θ = F 0(j)0β + Fi 1(j , s)Δ(j , s)

I baseline component F 0(j) characterizing the background secular trend in the absence of intervention

I a time-dependent intervention component Fi 1(j , s) = I[j≥s]

I β is the parameter encoding the secular trend

I Δ(j , s) is the change in the mean outcome at period j due to sequence s

I General representation of outcome model

g[µijk (s)] = F 0(j)0β + Fi 1(j , s)Δ(j , s)+ Rik (j , s)0αi .| {z } | {z } | {z }

secular trend intervention effect heterogeneity

I Useful for conceptualizing model elements

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Outcome Model - Cont’d

I Yijk (s) is then assumed to follow a parametric distribution with mean µijk (s) and variance as a function of µijk (s)

I Continuous, normally distributed outcome, obtain the linear mixed model (LMM)

Yijk (s) = F 0(j)0β + Fi 1(j , s)Δ(j , s) + Rik (j , s)0αi + �ijk

αi ∼ f (αi ; Θ), �ijk ∼ N(0, σ� 2)

I Heterogeneity parameter αi induces within-cluster correlations, or intraclass correlation coefficients (ICCs)

I Equate Yijk = Yijk (s), if cluster i receives sequence s

I Current literature has focused on a continuous outcome, review existing linear mixed model variants as special cases of the general representation

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3. Modeling Considerations &

Implications

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The Hussey and Hughes (HH) Model

I Yijk = µ+ βj + δXij + αi + �ijk6

I µ grand mean, βj is the j th period effect (β1 = 0)

I Xij intervention indicator, δ the intervention effect

I αi ∼ N(0, τα), �ijk ∼ N(0, σ� )2 2 independent of αi

I Secular trend F 0(j)0β = µ + β2I[j=2] + . . . + βJ I[j=J]

I Intervention effect, Δ(j , s) = δ, does not depend on the time interval during which the intervention was initiated

I Heterogeneity, Rik (j , s)0αi = αi , captures the cluster-specifc departure from the average but is assumed to be homogeneous across time periods and intervention sequences

6Hussey MA, Hughes JP (2007) Design and analysis of stepped wedge cluster randomized trials. Contemp Clin Trials 16\57

The Hussey and Hughes Model - Cont’d I Analogous to parallel CRT, the single cluster random effect

postulates a simple exchangeable correlation structure

I Common ICC: ρ = τ2/(τ2 + σ2 �αα )

I Sample size calculation

(σ2 tot/N)IJλ1λ2 var(δ̂) = ,

(U2 + IJU − JW − IV )λ2 − (U2 − IV )λ1

IPI J J I

σ2 = τα 2 + σ�

2, U = =1 Xij , W = =1( =1 Xij )2

tot i=1 j j iP P P

and I JV = i=1( j=1 Xij )2P P

are design constants

I λ1 = 1 − ρ, λ2 = 1 + (JN − 1)ρ eigenvalues of corr structure

I Many subsequent development based on the HH model

I limN→∞ var(δ̂) = 0, but most widely used7

7Taljaard M et al. (2016) Substantial risks associated with few clusters in cluster randomized and stepped wedge designs. Clinical Trials 17\57

Modeling Secular Trend

I Generally, F 0(j)0β = β1B1(j) + . . . + βpBp(j)

I F 0(j) = (B1(j), . . . , Bp(j))0 p-dimensional basis function

I Alternative approaches include linear specifcation F 0(j) = (1, j)0 or polynomial specifcation etc

I From an effciency perspective, favor the dimension of β controlled, except that

I var(δ̂) invariant to time parameterization as long�PI 0I i=1 Xi1, . . . , i=1 XiJ

�Play in the column space of

F 0 = (F 0(1), . . . , F 0(J))08— balanced allocation

I From a bias perspective, natural to consider a nonparametric representation of F 0(j)0β (as in HH model)

8Grantham KL et al. (2019) Time parameterizations in cluster randomized trial planning. Am Stat. 18\57

Modeling the Intervention Effect (a)

I Δ(j , s) = δ is a constant/averaged intervention effect

I Easy to work with, especially in the design stage

I Does not allow the strengthening or weakening of effect over time

19\57

Modeling the Intervention Effect (b) I Δ(j , s) can depend on period j and sequence s9

I Linear time-on-treatment effect

Δ(j , s) = δ0 + δ1(j − s), or, Δ(j , s) = δ(j − s + 1)

I Considers strengthening or weakening of effect over time

I Model in analyzing longitudinal parallel CRTs 9Hughes JP et al. (2015) Current issues in the design and analysis of

stepped wedge trials. Contemp Clin Trials 20\57

Modeling the Intervention Effect (c)

I Delayed treatment effect 9

Δ(j , s) = δπ0I[j=s] + δI[j>s],

I General delayed treatment effect

Δ(j , s) = δπ0I[j=s] + δπ1I[j=s+1] + . . . + δπJ−sI[j=J].

I Prior knowledge or assumptions on πj−s (πj−s = 0 if j < s) 21\57

Modeling the Intervention Effect (d)

I General time-on-treatment effect9

Δ(j , s) = δj−s = δ0I[j=s] + δ1I[j=s+1] + . . . + δJ−sI[j=J].

I Interpretable global tests

I H0: δ0 = δ1 = . . . = δJ−2 = 0 (no intervention effect) I H0: δ0 = δ1 = . . . = δJ−2 (constant intervention effect) I H0: δ1 − δ0 = δ2 − δ1 = . . . (linear time-on-treatment)

22\57

Preclude for Modeling Heterogeneity

I There has been extensive discussions of alternative strategies for modeling the random-effects structure in stepped wedge trials

I Centered on extensions to the Hussey and Hughes model

I constant intervention effect

I categorical time parameterization

I review variants of random-effects structures by assuming a linear link, categorical secular trend (with one exception) as well as a time-invariant intervention effect

23\57

Modeling Heterogeneity in CS Designs I Example extensions to the Hussey and Hughes model

cross-sectional designs; all models assume a continuous outcome and an identity link function10

10Li F. et al (2020) Mixed-effects models for the design and analysis of stepped wedge cluster randomized trials: An overview. Stat Methods Med Res 24\57

Nested Exchangeable Correlation Model

Yijk = µ + βj + δXij + αi + γij + �ijk

I γij ∼ ( , τ )γ N 0 2 , the random cluster-by-time interaction

I 0R j ( ) α α γs +=ik i i ij, , therefore allows the deviation from

the group average to be both cluster-specifc and period-specifc

I distinguishes between within-period ICC and between-period ICC11

�γ

�

αα

γ

γ

α α

= (τ2 + τ 2)/(τ2 + τ2 + σ2

= τ2/(τ2 + τ2 + σ2corr[Yijk (s), Yilm(s)] = ρw ), j = l

j = l ,ρb ),

(

6

11Hooper R et al (2016) Sample size calculation for stepped wedge and other longitudinal cluster randomised trials. Stat Med 25\57

Nested Exchangeable Correlation Model - Cont’d

I Sample size calculation takes into account both ICCs

I Same form of variance with HH model, except that

σ2 tot = τ2 + τ2 + σ2

�γα

λ1 = 1 + (N − 1)ρw − Nρb

λ2 = 1 + (N − 1)ρw + N(J − 1)ρb

I Hooper/Girling model (CAC and CMC) 11,12

I lim →∞ var(δ̂) = 0 N 612Girling AJ and Hemming K (2016) Statistical effciency and optimal

design for stepped cluster studies under linear mixed effects models. Stat Med 26\57

Exponential Decay Model

Yijk = µ + βj + δXij + γij + �ijk 13

I Heterogeneity term Rik (j , s)0αi = γij I γi = (γi1, . . . , γiJ )

0 ∼ N(0, τγ 2M) ⎛ ⎞ ⎟⎟⎟⎠

2 J−11 r0r r0r . . . r0r r0r 1 r0r . . . r0r J−2

. . . ... . . . ... . . . J−1 J−2 J−3r0r r0r r0r . . . 1

⎜⎜⎜⎝M =

I Allows between-period ICC to decay exponentially ( ρw = τγ

2/(τγ 2 + σ� 2), j = l

corr[Yijk (s), Yilm(s)] = ρb,|j−l| = ρwr |j−l|, j = l . 6

13Kasza J et al (2018). Impact of non-uniform correlation structure on sample size and power in multiple-period cluster randomised trials. Stat Methods Med Res 27\57

Exponential Decay Model - Cont’d

I Exponential decay and nested exchangeable correlation models do not have a clear nesting relationship

I Both can reduce to the HH model (ρb = ρw or r = 1)

I var(δ̂) does NOT exist in closed form

I From the design perspective, estimated sample size can go either direction when the incorrect model is assumed

I From the analysis perspective, omitting the decay parameter might lead to an inflated Type I error rate14

I The version of continuous-time correlation decay model (Grantham et al. 2020, Stat Med)

14Kasza J and Forbes AB (2018). Inference for the treatment effect in multiple-period cluster randomised trials when random effect correlation structure is misspecifed Stat Methods Med Res 28\57

Random Intervention Model

Yijk = µ + βj + (δ + νi )Xij + αi + �ijk ,

where � � �� � � �� αi 0 τα

2 σαν∼ N , ,τ 2νi 0 σαν ν

I random cluster-by-treatment interaction (Hughes et al. 2015)9

I heterogeneity term Rik (j , s)0αi = αi + νi I[j≥s]

I intervention-condition-specific correlation structures(treatment also affects variance components)

I careful on alternative parameterization to avoid strong assumptions15

15Hemming K et al (2018) Modeling clustering and treatment effect heterogeneity in parallel and stepped-wedge cluster randomized trials. Stat Med 29\57

Random Coeffcient Model

Yijk = µ + (β + ξi )Tj + δXij + αi + �ijk .

I Tj = j to represent the linear time basis function

I β as the fxed time slope and ξi as the random slope � � �� � � �� τ 2αi 0 α σαξ∼ N , .

ξi 0 σαξ τξ 2

I heterogeneity term Rik (j , s)0αi = αi + jξi

I used in longitudinal parallel CRTs16

I ICC structure τα

2 + (j + l)σαξ + jlτ 2

corr[Yijk (s), Yilm(s)] = q q ξ ,

τ 2 + σ2 τ 2 + σ2 α + 2jσαξ + j2τ 2

� α + 2lσαξ + l2τ 2 �ξ ξ

16Murray DM et al (1998). Analysis of data from group-randomized trials with repeat observations on the same groups. Stat Med 30\57

Random Coeffcient Model - Cont’d I Even this basic form remains to be studied more

I Could imply unique correlation structure opposed to exponential decay

(a) Zero covariance σαξ = 0

0.034 0.04 0.046 0.051 0.056

0.04 0.053 0.065 0.076 0.086

0.046 0.065 0.083 0.1 0.115

0.051 0.076 0.1 0.122 0.142

0.056 0.086 0.115 0.142 0.1685

4

3

2

1

1 2 3 4 5Period

Period

(b) Positive covariance σαξ = 0.5

0.059 0.077 0.093 0.108 0.122

0.077 0.1 0.122 0.141 0.159

0.093 0.122 0.148 0.172 0.194

0.108 0.141 0.172 0.2 0.225

0.122 0.159 0.194 0.225 0.2545

4

3

2

1

1 2 3 4 5Period

Period

31\57

Modeling Heterogeneity in CC Designs I Example extensions to the Hussey and Hughes model

closed-cohort designs; all models assume a continuous outcome and an identity link function17

17Li F. et al (2020) Mixed-effects models for the design and analysis of stepped wedge cluster randomized trials: An overview. Stat Methods Med Res 32\57

Basic Model Extending Hussey and Hughes

Yijk = µ + βj + δXij + αi + φik + �ijk 18 (1)

I φik ∼ N(0, τ2 φ) random effect for the repeated measures

I heterogeneity term Rik (j , s)0αi = αi + φik

I implies a nested exchangeable correlation structure

� φ

�

α

φ

)/(τ 2 + τ2 + σ2

+ τ2 + σ2 = m

(α

= (τ2 α

2τ= α/(τ2

2τ+ φ ), kρa = m corr[Yijk (s), Yilm(s)] =

k ρd ),

6

I closed-form variance shares the same form of HH variance (with changes in total variance and eigenvalues)

I limN→∞ var(δ̂) = 0!! 18Baio G et al. (2015) Sample size calculation for a stepped wedge trial.

Trials 33\57

Block Exchangeable Correlation Model

Yijk = µ + βj + δXij + αi + γij + φik + �ijk ,

I φik ∼ N(0, τ2)φ random effect for the repeated measures

I Hooper/Girling model

I heterogeneity Rik (j , s)0αi = αi + γij + φik

I correlation structure with three ICCs I within-period and between-period ICC different individuals

I (between-period) within-individual ICC for repeated measures (ρa)

I Same form of variance with HH model, except that 2 2 2 + τ 2

φ + σ2σ = τ + τtot ,α γ �

λ1 = 1 + (N − 1)(ρw − ρb) − ρa, λ2 = 1 + (N − 1)ρw + (J − 1)(N − 1)ρb + (J − 1)ρa.

I limN→∞ var(δ̂) = 0 6 34\57

Block Exchangeable Correlation Model - Cont’d

I Common choice of models in closed-cohort designs

I Closed-form variance allows us to confirm19

I within-period correlation ρw ↑, power ↓ (traditional ICC)

I between-period correlation ρb ↑, power ↑ I within-individual correlation ρa ↑, power ↑

I Only ρw is mostly likely to be found in the literature, few published ρb, ρa

I Conservatively small values for ρb, ρa will NOT underpower the study

19Li F et al. (2018) Sample size determination for GEE analyses of stepped wedge cluster randomized trials. Biometrics 35\57

Proportional Decay Model

Yijk = µ + βj + δXij + γij + �ijk ,

I Mimicking the exponential decay model with Rik (j , s)0αi = γij

I Further assume a similar autoregressive structure for residual errors of the k th person in cluster i as

�ik = (�i1k , . . . , �iJk )0 ∼ N(0, σ� 2M), �ik ⊥ �im, k = m 6

I Implies a proportional decay correlation structure ⎧ ⎪ρw = τγ 2/(τγ

2 + σ� 2), j = l, k = m,⎨

|j−l|corr[Yijk (s), Yilm(s)] = ρa,|j−l| = r , j = l , k = m,⎪⎩ |j−l|ρb,|j−l| = ρwr , j = l , k = m,

66

6 6

I ρw and ρb,|j−l| are the within-period and between-period ICCs

I ρa,|j−l| the within-individual ICC that decays exponentially 36\57

Proportional Decay Model - Cont’d I Originally studied under marginal model20

I Assumption: same decay rate r applies to both the within-individual ICC and the between-period ICC for different individuals

I Separability of correlation matrix allow us to obtain

(σ2 2){1 + (N − 1)ρw}tot/N)I(1 − r var(δ̂) = ,

(IU − W )(1 + r2) − 2(IP − Q)r

wherePI J J I U = i=1 j=1 Xij , W = j=1( i=1 Xij )

2,P P PPI J−1 J−1 I IP = i=1 j=1 XijXi,j+1, Q = j=1 ( i=1 Xij )( i=1 Xi,j+1)

P P P Pare new design constants

I Parabolic relationship between var(δ̂) and decay

20Li F (2020). Design and analysis considerations for cohort stepped wedge cluster randomized trials with a decay correlation structure. Stat Med 37\57

Proportional Decay Model - Cont’d I Sample size calculation may be sensitive to correlation

assumptions, and can go either direction 20

I recall nested exchangeable versus exponential decay (cross-sectional)

I block exchangeable (BE) versus proportional decay (PD) (closed-cohort)

I contour of varPD(δ̂)/varBE(δ̂) I common ρw = 0.03

I r = 0.5 from PD

I x-axis ρb from BE

I y-axis ρa from BE

20Li F (2020). Design and analysis considerations for cohort stepped wedge cluster randomized trials with a decay correlation structure. Stat Med 38\57

Random Intervention Model

Yijk = µ + βj + (δ + νi )Xij + γij + φik + �ijk ,21

I φik ∼ N(0, τ2) φ

for repeated measures , γij is the cluster-period-specifc random deviation from the group average, as in the exponential decay model

I νi is the cluster-specifc random intervention effect

I Rik (j , s)0αi = γij + φik + νi I[j≥s]

I implies eight different ICC parameters

I generalizes exponential decay and random intervention models under CS design, but does not nest proportional decay model

21Kasza J et al. (2019) Information content of stepped wedge designs when treatment effect heterogeneity and/or implementation periods are present. Stat Med 39\57

Modeling Heterogeneity in OC Designs

I Open-cohort (OC) design can be considered as a mix of a cross-sectional design and a closed-cohort design

I Nij individuals included during period j in cluster i

I Exists an overlapping number � �0 ≤ ni (j , l) ≤ min{Nij , Nil }

of individuals for period j and period l in cluster i , depending on the degree of cohort openness

I Notation generalizes that of the previous two designs

I cross-sectional: ni (j , l) = 0 for all j and l (maximum degree of openness)

I closed-cohort: ni (j , l) = Nij = Nil for all j and l (minimum degree of openness).

40\57

Modeling Heterogeneity in OC Designs - Cont’d

I In principle, the models developed for the closed-cohort design can still be used to represent the outcome trajectory in the open-cohort design

I Caveat is the repeated measures are only available for individuals included in more than one period

I For example, the block exchangeable model still applies

Yijk = µ + βj + δXij + αi + γij + φik + �ijk ,

I The implied within-cluster correlation matrix is neither nested exchangeable nor block exchangeable, but becomes a blend of these two

41\57

Blended Block Correlation Structure

Each block represents a given cluster-period or between two cluster-periods, and J = 3. In the open-cohort design, we assume only one individual is followed through all periods, and a new individual will be supplemented in each period.

I All three matrices have the same diagonal block

42\57

Blended Block Correlation Structure Each block represents a given cluster-period or between two cluster-periods, and J = 3. In the open-cohort design, we assume only one individual is followed through all periods, and a new individual will be supplemented in each period.

I All three matrices have the same diagonal block

I Difference in off-diagonal blocks determined by overlapping number of individuals and hence degree of cohort openness

43\57

Design Considerations

I Assuming same cluster-period sizes and constant attrition rate, Kasza et al. derived the closed-form variance for power calculation 22

I The attrition rate refects the degree of openness — represents continuum between cross-sectional and closed-cohort designs

I facilitates effciency comparisons between these two designs

I closed-cohort design is usually at least as efficient as the cross-sectional design as long as ρa ≥ ρb

I the re verse when ρa < ρb (not plausible under mixed models)

I Extensions to the correlation decay22

22Kasza J et al. (2020) Sample size and power calculations for open cohort longitudinal cluster randomized trials. Stat Med 44\57

Considerations for Binary Outcomes

I Not as many article on binary outcomes, which are nonetheless common as primary endpoints

I From design perspective

I Zhou et al.23 provided a maximum likelihood approach for power calculation with binary outcomes

I extending the HH model to estimate risk difference

I SAS and R package forthcoming swdpwr I can be accessed via https://publichealth.yale. edu/cmips/research/software/swdpwr/

I Important message is that linear mixed model approximation may not be accurate for power calculation with binary outcomes

23Zhou X et al. (2020) A maximum likelihood approach to power calculations for stepped wedge designs of binary outcomes. Biostatistics 45\57

Considerations for Binary Outcomes - Cont’d I Limited investigations on generalized linear mixed models

with more complex random-effects structure

I An exception is Thompson et al.24, compared

I Hussey and Hughes model

I Nested exchangeable model (Hooper/Girling model)

I Random intervention model

I The logistic nested exchangeable correlation model

logit(µij ) = µ+βj +δXij +αi +γij , αi ∼ N(0, τα 2), γij ∼ N(0, τγ

2)

had more robust performance in terms of bias and type I error rates across a number of data generating processes

I Careful on the interpretation of δ 24Thompson et al. (2017) Bias and inference from misspecifed

mixed-effect models in stepped wedge trial analysis. Stat Med 46\57

Estimation and Inference

I Fitting variants of mixed effects models have become standard in common software

I proc mixed, glimmix or hpmixed (SAS)

I nlme or lme4 (R)

I Flexible choice of readily-available complex random effects structure with linear mixed model compared to generalized linear mixed model

I provide intervention effect parameter estimate

I estimate variance component, but need additional step to compute ICC (simple for continuous outcomes, not as much for binary)

47\57

Estimation and Inference - Cont’d

I Permutation inference has gained traction for accurate type I error rate control

I General idea is to obtain the reference distribution of a given test statistic by permuting the intervention sequences across clusters

I requires exchangeability across permuted intervention sequences under the null

I Other recent permutation methods discussed in Li et al.25

25Li F. et al (2020) Mixed-effects models for the design and analysis of stepped wedge cluster randomized trials: An overview. Stat Methods Med Res 48\57

4. Concluding Remarks

49\57

On Mixed Model Variants

I Review variants of mixed models for stepped wedge cluster randomized trials under a unifed perspective

g[µijk (s)] = F 0(j)0β + Fi 1(j , s)Δ(j , s)+ Rik (j , s)0αi .| {z } | {z } | {z }

secular trend intervention effect heterogeneity

I Majority of models assumed categorical time effect and a scalar intervention effect (convenient for sample size estimation)

I Current literature devoted to continuous outcomes and variations of random-effects structure

I Relatively limited literature on binary or count outcomes

50\57

Choices of Models

I Generally a diffcult question with no uniform solution

I Can depend on the accuracy of characterizing the outcome trajectories

I Consistent choice of models in design and analysis stage through pre-specifcation

I Design stage – prior information/pilot data/sensitivity analysis

I Analysis stage – I robust analysis

I information criteria?26

I open question

26Murray DM et al (1998). Analysis of data from group-randomized trials with repeat observations on the same groups. Stat Med 51\57

Alternatives: Marginal Models

I Marginal models I separate mean and correlation models

I population-averaged interpretation

I inference robust to correlation specification (# cluster I large)

I Recent literature studying sample size and finite-sample behaviour of GEE estimators, e.g27

I Directly estimating correlations instead of variance components

I Software available, but more need to be developed

27Li F et al. (2018) Sample size determination for GEE analyses of stepped wedge cluster randomized trials. Biometrics 52\57

Revisit CONSORT extension to SW-CRTs

I CONSORT item 7a: Sample size28

I Extension for SW-CRTs – . . . Method of calculation and relevant parameters with sufficient detail so the calculation can be replicated. Assumptions made about correlations between outcomes of participants from the same cluster.

I CONSORT item 12a & b: Statistical methods

I Extension for SW-CRTs – . . . Statistical methods used to compare treatment conditions for primary and secondary outcomes including how time effects, clustering, and repeated measures were taken into account..

I Important to explicitly describe model variants

28Hemming K (2018) Reporting of stepped wedge cluster randomised trials: Extension of the CONSORT 2010 statement with explanation and elaboration. BMJ 53\57

More on CONSORT extension to SW-CRTs

I CONSORT item 17a: Outcomes and estimation

I Extension for SW-CRTs – For each primary and secondary outcome, results for each treatment condition, and the estimated effect size and its precision; any correlations (or covariances) and time effects estimated in the analysis outcome

I With either choice of models, reporting ICC or variance components is highly recommended as this can be informative for the planning of future trials

I In particular for the correlation decay and random intervention models, to facilitate the design of trials based on these more recent extensions

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Thank you for listening!

I questions are welcome either during the webinar or fan.f.li@yale.edu

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Back up slide: Other Important Aspects Not Mentioned

I Using baseline covariates?

I constrained randomization (design)

I improve power (analysis)

I Multiple layers of clustering

I Addressing “missing data"

I complete designs, missing outcomes (e.g. closed-cohort)

I incomplete designs

I Other interesting and important questions to be solved

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